\(\int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx\) [174]

Optimal result

Integrand size = 20, antiderivative size = 358 \[ \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=-\frac {2 d \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {a+b \text {arccosh}(c x)}}-\frac {2 e x^2 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {e e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {e e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3} \] Output:

-2*d*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c/(a+b*arccosh(c*x))^(1/2)-2*e*x^2*(c*x 
-1)^(1/2)*(c*x+1)^(1/2)/b/c/(a+b*arccosh(c*x))^(1/2)+d*exp(a/b)*Pi^(1/2)*e 
rf((a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c+1/4*e*exp(a/b)*Pi^(1/2)*erf 
((a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^3+1/4*e*exp(3*a/b)*3^(1/2)*Pi 
^(1/2)*erf(3^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^3+d*Pi^(1/2 
)*erfi((a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c/exp(a/b)+1/4*e*Pi^(1/2) 
*erfi((a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^3/exp(a/b)+1/4*e*3^(1/2) 
*Pi^(1/2)*erfi(3^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^3/exp(3 
*a/b)
 

Mathematica [A] (warning: unable to verify)

Time = 1.24 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.75 \[ \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\frac {e^{-\frac {3 a}{b}} \left (-\left (\left (4 c^2 d+e\right ) e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c x)\right )\right )+\sqrt {3} e \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+\left (4 c^2 d+e\right ) e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )-e^{\frac {3 a}{b}} \left (2 \left (4 c^2 d+e\right ) \sqrt {\frac {-1+c x}{1+c x}} (1+c x)+\sqrt {3} e e^{\frac {3 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+2 e \sinh (3 \text {arccosh}(c x))\right )\right )}{4 b c^3 \sqrt {a+b \text {arccosh}(c x)}} \] Input:

Integrate[(d + e*x^2)/(a + b*ArcCosh[c*x])^(3/2),x]
 

Output:

(-((4*c^2*d + e)*E^((4*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, a/b + Arc 
Cosh[c*x]]) + Sqrt[3]*e*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-3*(a 
+ b*ArcCosh[c*x]))/b] + (4*c^2*d + e)*E^((2*a)/b)*Sqrt[-((a + b*ArcCosh[c* 
x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c*x])/b)] - E^((3*a)/b)*(2*(4*c^2*d + 
e)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) + Sqrt[3]*e*E^((3*a)/b)*Sqrt[a/b + 
 ArcCosh[c*x]]*Gamma[1/2, (3*(a + b*ArcCosh[c*x]))/b] + 2*e*Sinh[3*ArcCosh 
[c*x]]))/(4*b*c^3*E^((3*a)/b)*Sqrt[a + b*ArcCosh[c*x]])
 

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6324, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx\)

\(\Big \downarrow \) 6324

\(\displaystyle \int \left (\frac {d}{(a+b \text {arccosh}(c x))^{3/2}}+\frac {e x^2}{(a+b \text {arccosh}(c x))^{3/2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {\pi } e e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {\sqrt {3 \pi } e e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {\sqrt {\pi } e e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {\sqrt {3 \pi } e e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {\sqrt {\pi } d e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {\sqrt {\pi } d e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}-\frac {2 d \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}-\frac {2 e x^2 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}\)

Input:

Int[(d + e*x^2)/(a + b*ArcCosh[c*x])^(3/2),x]
 

Output:

(-2*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*Sqrt[a + b*ArcCosh[c*x]]) - (2*e* 
x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*Sqrt[a + b*ArcCosh[c*x]]) + (d*E^(a 
/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(b^(3/2)*c) + (e*E^(a/ 
b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*b^(3/2)*c^3) + (e*E^ 
((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(4*b 
^(3/2)*c^3) + (d*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(b^(3/2) 
*c*E^(a/b)) + (e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*b^(3/ 
2)*c^3*E^(a/b)) + (e*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sq 
rt[b]])/(4*b^(3/2)*c^3*E^((3*a)/b))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6324

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (d + e*x^2)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && 
(p > 0 || IGtQ[n, 0])
 

Maple [F]

\[\int \frac {e \,x^{2}+d}{\left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{\frac {3}{2}}}d x\]

Input:

int((e*x^2+d)/(a+b*arccosh(c*x))^(3/2),x)
 

Output:

int((e*x^2+d)/(a+b*arccosh(c*x))^(3/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((e*x^2+d)/(a+b*arccosh(c*x))^(3/2),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

\[ \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\int \frac {d + e x^{2}}{\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((e*x**2+d)/(a+b*acosh(c*x))**(3/2),x)
 

Output:

Integral((d + e*x**2)/(a + b*acosh(c*x))**(3/2), x)
 

Maxima [F]

\[ \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\int { \frac {e x^{2} + d}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((e*x^2+d)/(a+b*arccosh(c*x))^(3/2),x, algorithm="maxima")
 

Output:

integrate((e*x^2 + d)/(b*arccosh(c*x) + a)^(3/2), x)
 

Giac [F]

\[ \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\int { \frac {e x^{2} + d}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((e*x^2+d)/(a+b*arccosh(c*x))^(3/2),x, algorithm="giac")
 

Output:

integrate((e*x^2 + d)/(b*arccosh(c*x) + a)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\int \frac {e\,x^2+d}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{3/2}} \,d x \] Input:

int((d + e*x^2)/(a + b*acosh(c*x))^(3/2),x)
 

Output:

int((d + e*x^2)/(a + b*acosh(c*x))^(3/2), x)
 

Reduce [F]

\[ \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx =\text {Too large to display} \] Input:

int((e*x^2+d)/(a+b*acosh(c*x))^(3/2),x)
 

Output:

( - acosh(c*x)*int((sqrt(acosh(c*x)*b + a)*x**4)/(acosh(c*x)**2*b**2*c**2* 
x**2 - acosh(c*x)**2*b**2 + 2*acosh(c*x)*a*b*c**2*x**2 - 2*acosh(c*x)*a*b 
+ a**2*c**2*x**2 - a**2),x)*b**2*c**5*d + 2*acosh(c*x)*int((sqrt(acosh(c*x 
)*b + a)*x**4)/(acosh(c*x)**2*b**2*c**2*x**2 - acosh(c*x)**2*b**2 + 2*acos 
h(c*x)*a*b*c**2*x**2 - 2*acosh(c*x)*a*b + a**2*c**2*x**2 - a**2),x)*b**2*c 
**3*e + acosh(c*x)*int((sqrt(acosh(c*x)*b + a)*x**2)/(acosh(c*x)**2*b**2*c 
**2*x**2 - acosh(c*x)**2*b**2 + 2*acosh(c*x)*a*b*c**2*x**2 - 2*acosh(c*x)* 
a*b + a**2*c**2*x**2 - a**2),x)*b**2*c**3*d - 2*acosh(c*x)*int((sqrt(acosh 
(c*x)*b + a)*x**2)/(acosh(c*x)**2*b**2*c**2*x**2 - acosh(c*x)**2*b**2 + 2* 
acosh(c*x)*a*b*c**2*x**2 - 2*acosh(c*x)*a*b + a**2*c**2*x**2 - a**2),x)*b* 
*2*c*e + 6*acosh(c*x)*int((sqrt(c*x + 1)*sqrt(c*x - 1)*sqrt(acosh(c*x)*b + 
 a)*acosh(c*x)*x**3)/(acosh(c*x)**2*b**2*c**2*x**2 - acosh(c*x)**2*b**2 + 
2*acosh(c*x)*a*b*c**2*x**2 - 2*acosh(c*x)*a*b + a**2*c**2*x**2 - a**2),x)* 
b**2*c**4*d + 6*acosh(c*x)*int((sqrt(c*x + 1)*sqrt(c*x - 1)*sqrt(acosh(c*x 
)*b + a)*x**3)/(acosh(c*x)**2*b**2*c**2*x**2 - acosh(c*x)**2*b**2 + 2*acos 
h(c*x)*a*b*c**2*x**2 - 2*acosh(c*x)*a*b + a**2*c**2*x**2 - a**2),x)*a*b*c* 
*4*d - 2*sqrt(c*x + 1)*sqrt(c*x - 1)*sqrt(acosh(c*x)*b + a)*c**2*d*x**2 - 
4*sqrt(c*x + 1)*sqrt(c*x - 1)*sqrt(acosh(c*x)*b + a)*d - int((sqrt(acosh(c 
*x)*b + a)*x**4)/(acosh(c*x)**2*b**2*c**2*x**2 - acosh(c*x)**2*b**2 + 2*ac 
osh(c*x)*a*b*c**2*x**2 - 2*acosh(c*x)*a*b + a**2*c**2*x**2 - a**2),x)*a...